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Covering a Polyomino-Shaped Stain with Non-Overlapping Identical Stickers

Authors Keigo Oka , Naoki Inaba, Akira Iino

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LIPIcs.FUN.2026.37.pdf
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  • 21 pages

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Problems, reductions and completeness
Keywords
  • polyomino
  • covering
  • NP-completeness

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Abstract

You find a stain on the wall and decide to cover it with non-overlapping stickers of a single identical shape (rotation and reflection are allowed). Is it possible to find a sticker shape that fails to cover the stain? In this paper, we consider this problem under polyomino constraints and complete the classification of always-coverable stain shapes (polyominoes). We provide proofs for the maximal always-coverable polyominoes and construct concrete counterexamples for the minimal not always-coverable ones, demonstrating that such cases exist even among hole-free polyominoes. This classification consequently yields an algorithm to determine the always-coverability of any given stain. We also show that the problem of determining whether a given sticker can cover a given stain is NP-complete, even though exact cover is not demanded. This result extends to the 1D case where the connectivity requirement is removed. As an illustration of the problem complexity, for a specific hexomino (6-cell) stain, the smallest sticker found in our search that avoids covering it has, although not proven minimum, a bounding box of 325 × 325.

Cite As Get BibTex

Keigo Oka, Naoki Inaba, and Akira Iino. Covering a Polyomino-Shaped Stain with Non-Overlapping Identical Stickers. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 37:1-37:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.37

Author Details

Keigo Oka
  • Google, Tokyo, Japan
Naoki Inaba
  • Puzzle creator, Japan
Akira Iino
  • Nippon Hyoron Sha, Co., Ltd., Toshima, Japan

Funding

  • Oka, Keigo: Work done in a personal capacity.

Supplementary Materials

References

  1. Wallace C Babcock. Intermodulation interference in radio systems frequency of occurrence and control by channel selection. The Bell System Technical Journal, 32(1):63-73, 1953. Google Scholar
  2. D Beauquier and M Nivat. On translating one polyomino to tile the plane. Discrete Comput. Geom., 6(4):575-592, December 1991. URL: https://doi.org/10.1007/BF02574705.
  3. Siddhartha Bhattacharya. Periodicity and decidability of tilings of 𝐙². American Journal of Mathematics, 142(1):255-266, 2020. Google Scholar
  4. Marc Demange and Dominique de Werra. On some coloring problems in grids. Theoretical Computer Science, 472:9-27, 2013. URL: https://doi.org/10.1016/j.tcs.2012.10.046.
  5. Konstantinos Drakakis. A review of the available construction methods for Golomb rulers. Adv. Math. Commun., 3(3):235-250, 2009. URL: https://doi.org/10.3934/amc.2009.3.235.
  6. Paul Erdos and Pál Turán. On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc, 16(4):212-215, 1941. Google Scholar
  7. Ian Gambini and Laurent Vuillon. An algorithm for deciding if a polyomino tiles the plane. RAIRO - Theoretical Informatics and Applications, 41(2):147-155, 2007. URL: https://doi.org/10.1051/ita:2007012.
  8. Martin Gardner. More about the shapes that can be made with complex dominoes. Scientific American, 203(5):186-+, 1960. Google Scholar
  9. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  10. Solomon W Golomb. Polyominoes: puzzles, patterns, problems, and packings, volume 111. Princeton University Press, 1996. Google Scholar
  11. Takashi Horiyama, Takehiro Ito, Keita Nakatsuka, Akira Suzuki, and Ryuhei Uehara. Complexity of tiling a polygon with trominoes or bars. Discrete & Computational Geometry, 58:686-704, 2017. URL: https://doi.org/10.1007/s00454-017-9884-9.
  12. Naoki Inaba. 10個の点. http://inabapuzzle.com/hirameki/suuri_4.html, 2008. (in Japanese).
  13. Naoki Inaba. 1×5が覆えない http://inabapuzzle.com/hirameki/suuri_7.html, 2011. (in Japanese).
  14. Peter J. M. Laarhoven and Emile H. L. Aarts. Simulated Annealing: Theory and Applications, volume 37 of Mathematics and Its Applications. Springer, 1987. URL: https://doi.org/10.1007/978-94-015-7744-1.
  15. Cristopher Moore and John Michael Robson. Hard tiling problems with simple tiles. Discrete & Computational Geometry, 26(4):573-590, 2001. URL: https://doi.org/10.1007/s00454-001-0047-6.
  16. Yosuke Okayama, Masashi Kiyomi, and Ryuhei Uehara. On covering of any point configuration by disjoint unit disks. In Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Toronto, Ontario, Canada, August 10-12, 2011, 2011. URL: http://www.cccg.ca/proceedings/2011/papers/paper5.pdf.
  17. Nicolas Ollinger. Tiling the plane with a fixed number of polyominoes. In Language and Automata Theory and Applications, Lecture notes in computer science, pages 638-647. Springer, Berlin, 2009. URL: https://doi.org/10.1007/978-3-642-00982-2_54.
  18. S. Sidon. Ein Satz über trigonometrische Polynome und seine Anwendung in der Theorie der Fourier-Reihen. Mathematische Annalen, 106:536-539, 1932. Google Scholar
  19. David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. An aperiodic monotile. Comb. Theory, 4(1), 2024. URL: https://doi.org/10.5070/c64163843.
  20. Peter Winkler. Puzzled: Solutions and sources. Commun. ACM, 53(9):110, September 2010. Google Scholar
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